Conformal Group and Its Connection with an Indefinite Metric in Hilbert Space

It is shown that from any unitary representation of the 15-parameter conformal group, with the scalar product ($f_1,f_2$), another representation can be constructed in the same linear space with the indefinite metric ($f_1,R{f}_2$). $R$ is the operator which represents the transformation by reciprocal radii in that space. Its eigen-values are ±1. In the momentum space of the Klein-Gordon equation without rest mass, $R$ is essentially the Hankel transformation and its eigenfunctions are Laguerre's functions. The new representations solve the problem of representing the dilatations in Hilbert space and lead to a less singular quantization in field theory. The canonical $\delta$ function in momentum space is replaced by a Bessel function of order zero.